Birefringent Gravitational Waves and the Consistency Check of Inflation

TL;DR

The paper investigates how a gravitational Chern-Simons term, motivated by stringy leptogenesis, modifies inflationary gravitational waves. By deriving the modified equations of motion, the slow-roll effective potential, and the super-horizon power spectra for the two GW polarizations, the authors show that the amplitude is polarization-dependent and that the tensor-to-scalar ratio acquires a Θ^2 correction while the spectral index remains unchanged at leading order. They identify a scale-dependent divergence in the left-polarization mode, signaling a breakdown of linear perturbation and requiring non-linear treatment for sizable effects. The results establish a direct link between high-energy stringy dynamics and CMB observables, albeit with the caveat that observable signatures in the linear regime are suppressed unless non-linear effects are considered.

Abstract

In this work we show that the gravitational Chern-Simons term, aside from being a key ingredient in inflationary baryogenesis, modifies super-horizon gravitational waves produced during inflation. We compute the super-Hubble gravitational power spectrum in the slow-roll approximation and show that its overall amplitude is modified while its spectral index remains unchanged (at leading order in the slow-roll parameters). Then, we calculate the correction to the tensor to scalar ratio, T/S. We find a correction of T/S which is dependent on $\cal{N}$ (more precisely quadratic in ${\cal N}$), the parameter characterizing the amplitude of the Chern-Simons terms. In a stringy embedding of the leptogenesis mechanism, $\cal{N}$ is the ratio between the Planck scale and the fundamental string scale. Thus, in principle, we provide a direct probe of leptogenesis due to stringy dynamics in the Cosmic Microwave Background (CMB). However, we demonstrate that the corresponding correction of T/S is in fact very small and not observable in the regime where our calculations are valid. To obtain a sizable effect, we argue that a non-linear calculation is necessary.

Figures (1)

• Figure 1: Effective potential for the two states of polarization (solid line for the right polarization state and dashed line for the left polarization state). At $x=-1$ or $\eta =-8/(k\Theta )$, the effective potential $f_{_{\rm L}}(x)$ blows up. For $x>-1$, the slight difference between $f_{_{\rm L}}(x)$ and $f_{_{\rm R}}(x)$ mathematically originates from the term $\lambda ^s/[x(1-\lambda ^sx)]$ in $z_s"/z_s$ and, physically, from the phenomenon of birefringence. As $x\rightarrow 0$, the standard term $(2+3\epsilon)/x^2$ dominates. Since this term does not depend on the polarization state, one has $f{_{\rm R}}(x)\rightarrow f{_{\rm L}}(x)$.